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case, periods of two O's each may be annexed and the operation continued until any desired decimal order of the root is reached. If the number contains an odd number of decimal places, add a 0.

CASE II. TO FIND THE SQUARE ROOT OF A COMMON FRAC

TION.

RULE. Find the square roots of the numerator and denominator separately if both are perfect squares; if either is not, reduce the fraction to a decimal and find the square root of this.

Solution:

PROBLEMS.

√ 730.05=what? 1. √1,444-what? Ans. 38.

730.05)27.019+, Ans. 2. √118.81=what? Ans. 10.9.

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CHAPTER IX

GEOMETRICAL MAGNITUDES

A geometrical magnitude is a quantity that has either length, breadth, thickness or form.

There are four kinds of geometrical magnitudes: lines, surfaces, solids, and angles. A point has position, but not magnitude.

Some of the subdivisions of these magnitudes are as follows:

LINES.

A line is a magnitude that has length only.

A straight line is a line that does not change direction at any point.

A curved line is a line that changes direction at every point. A straight line is ordinarily called a right line, or simply a line; a curved line, a curve.

[For simplicity, the following definitions regarding surfaces and angles are here given:

A surface is a magnitude that has length and breadth, but not thickness.

A plane surface, or plane, is a surface in which a straight line, joining any two points in the surface, will wholly lie.

A plane angle, or an angle, as before defined, is the opening between two straight lines which meet at a point; as a magnitude it has form, but neither length, breadth, nor thickness. A right angle is an angle measured by 90°.]

Parallel lines are lines, in the same plane, which would not

meet however far either way both of them might be extended; as the lines AB, CD. (Fig. 2.)

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Perpendicular lines are straight lines at right angles to each other; as the lines AB, CD. (Fig. 3.)

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A vertical line is one that points toward the center of the earth. A plummet is ordinarily used to determine this direction.

A horizontal line is one at right angles to a vertical line. It is parallel to the horizon, or sea level, and may be determined by means of a plummet and carpenter's square or by a carpenter's level.

ANGLES.

An acute angle is one that is less than a.right angle; as the angle BAC. (Fig. 4.)

FIG. 4.

B

An obtuse angle is one that is greater than a right angle; as the angle CAB. (Fig. 5.)

FIG. 5.

When one straight line intersects another the four angles formed are named according to their relative positions, thus: Adjacent angles are the two angles on the same side of either line; as the angles AOC and COB are adjacent.

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Opposite angles are either pair of those which point in opposite directions; as AOD and COB.

When one straight line intersects two parallel lines the angles within the parallels on different sides of the intersecting line, but not adjacent, are called alternate angles; as AHG and HGD.

D

FIG. 7.

An interior angle is an angle formed inside of an inclosed

figure by the meeting of two of its sides; as ABC.

An exterior angle is an angle formed outside the figure by any side and the prolongation of an adjacent side; as aAB.

A

FIG. 8.

A reentering angle is one that points inward, as ADC; a salient angle, one that points outward, as ABC.

SURFACES.

A curved surface is a surface no part of which is a plane. A polygon is a plane surface bounded by straight lines. A regular polygon is one whose sides are equal.

Polygons are named according to the number of their sides; one of three sides is a triangle; of four, a quadrilateral; of five, a pentagon; of six a hexagon; and so on.

The following diagrams represent regular polygons:

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A diagonal of a polygon is a straight line joining any two

angles not adjacent.

The base is the side on which the polygon stands.

The altitude is the perpendicular distance from the base to the highest point, or one of the highest points, of the polygon. The perimeter of a polygon is its bounding line.

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